the connection: truth tables to functions
DESCRIPTION
abcF 0000 0011 0101 0111 1000 1011 1101 1110. OR. The Connection: Truth Tables to Functions. Condition that a is 0, b is 0, c is 1. Function F is true if any of these and-terms are true!. Sum-of-Products form. abcF 0000 0011 0101 0111 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/1.jpg)
The Connection: Truth Tables to Functions
a b c F
0 0 0 00 0 1 10 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
a b c a b c a b c
a b c a b c
Function F is true if any ofthese and-terms are true!
Condition that a is 0, b is 0, c is 1.
OR
Sum-of-Products form
F a b c a b c a b c a b c a b c ( ) ( ) ( ) ( ) ( )
![Page 2: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/2.jpg)
Minterm Shorthand
F a b c a b c a b c a b c a b cF
( ) ( ) ( ) ( ) ( )
( , , , , ) m + m + m + m + m
F = 1 2 3 5 6
m 1 2 3 5 6
a b c a b c a b c
a b c a b c
a b c
a b c a b c
= m0
= m1
= m2
= m3
= m4
= m5
= m6
= m7
Note: Binary ordering
A minterm has one literal for each input variable, either in its normal or complemented form.
A canonical sum-of-products form of an expression consists only of minterms OR’d together
a b c F
0 0 0 00 0 1 10 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
![Page 3: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/3.jpg)
Minterms of Different SizesTwo variables:
a b minterm0 0 a’b’ = m0
0 1 a’b = m1
1 0 a b’ = m2
1 1 a b = m3
Three variables:a b c minterm0 0 0 a’b’c’ = m0
0 0 1 a’b’c = m1
0 1 0 a’b c’ = m2
0 1 1 a’b c = m3
1 0 0 a b’c’ = m4
1 0 1 a b’c = m5
1 1 0 a b c’ = m6
1 1 1 a b c = m7
Four variables:
a b c d minterm0 0 0 0 a’b’c’d’ = m0
0 0 0 1 a’b’c’d = m1
0 0 1 0 a’b’c d’ = m2
0 0 1 1 a’b’c d = m3
0 1 0 0 a’b c’d’ = m4
0 1 0 1 a’b c’d = m5
0 1 1 0 a’b c d’ = m6
0 1 1 1 a’b c d = m7
1 0 0 0 a b’c’d’ = m8
1 0 0 1 a b’c’d = m9
1 0 1 0 a b’c d’ = m10
1 0 1 1 a b’c d = m11
1 1 0 0 a b c’d’ = m12
1 1 0 1 a b c’d = m13
1 1 1 0 a b c d’ = m14
1 1 1 1 a b c d = m15
![Page 4: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/4.jpg)
Sum-of-Products Minimization
F a b c a b c a b c a b c a b c ( ) ( ) ( ) ( ) ( )
F in canonical sum-of-products form (minterm form):
Use algebraic manipulation to make a simpler sum-of-products form
)()()()()()( cbacbacbacbacbacbaF
Use commutativity to reorder to group similar terms
))(())(())(( cbaabacccbaaF Use distributivity to factor out common terms)()()( cbbacbF
Use x’+x = 1 identity
Duplicate term - OK
We will find a better method (K-maps) later…
![Page 5: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/5.jpg)
Product-of-Sums from a Truth Table
A
0 0 0 0 1 1 1 1
B
0 0 1 1 0 0 1 1
C
0 1 0 1 0 1 0 1
F
0 0 0 1 1 1 1 1
F
1 1 1 0 0 0 0 0
CBACBACBAF
Use DeMorgan’s Law to re-express as product-of sums
Find an expressionfor F’ (the complement)
Complement both sides…
)()()( CBACBACBAF
CBACBACBAF
CBACBACBAF
![Page 6: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/6.jpg)
Maxterms
A
0 0 0 0 1 1 1 1
B
0 0 1 1 0 0 1 1
C
0 1 0 1 0 1 0 1
F
0 0 0 1 1 1 1 1
F
1 1 1 0 0 0 0 0
F A B C A B C A B C ( ) ( ) ( )
To find a Product-of-Sums form for a truth table Make one maxterm for each row in which the function is
zero For each maxterm, each variable appears once
In its complemented form if it is one in the row In its regular form if it is zero in the row
Maxterms
![Page 7: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/7.jpg)
Maxterm ShorthandProduct of Sums
F in canonical maxterm form:
A B C Maxterms
A + B + C = M7
A + B + C = M6
A + B + C = M5
A + B + C = M4
A + B + C = M3
A + B + C = M2
A + B + C = M1
A + B + C = M00 0 00 0 10 1 0
0 1 11 0 01 0 11 1 01 1 1
F A B C A B C A B CF M M MF
( ) ( ) ( )
0 1 2
M(0, 1, 2)
![Page 8: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/8.jpg)
Boolean operations and gates
Theorem: Any operation than can be represented by a truth table can be represented in Boolean algebra All truth tables can be made out of only and,
or, and not functions
![Page 9: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/9.jpg)
NAND/NOR expressionsAny expression can be made of and ANDs, ORs and NOTs
Thus, we can make any expression out of NANDs, NORs, and NOTs
So, we can make any expression out of just NANDs and NORs
X X
note: NANDs and NORs are easy to build with switches
We can make ANDs and ORs from NANDs and NORs and NOTs
We can make NOTs out of a single NAND gate
![Page 10: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/10.jpg)
NAND-only circuitsUsing DeMorgan’s Law
NORs can be made with NANDs!
We can make any Boolean expression out of only NAND Gates
NANDs can be made out of NORs!
We can make any Boolean expression out of only NOR Gates
![Page 11: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/11.jpg)
Sum-of-Products Circuits with NANDs Introduce Double Inverters
Sum-of-Productsworks well with NANDs
DeMorgan’sLaw
![Page 12: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/12.jpg)
Product-of-Sums Circuits with NORs Introduce Double Inverters
Product-of-Sumsworks well with NORs
DeMorgan’sLaw
![Page 13: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/13.jpg)
Converting General Circuits to NANDs
AB
D
CB
ACD
BD
Introduce Double Inverters to make NANDs:
Add inverters as needed to maintain correct polarity
Represent inverters with NANDs
![Page 14: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/14.jpg)
Seven-Segment ExampleA seven-segment display is used to display numbers
a
b
c
d
e
f g
a b c d e f
b c
a b d e g
a b c d g
b c f g
a c d f g
a c d e f g
a b c
a b c d e f g
a b c d f g
![Page 15: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/15.jpg)
Seven Segment Truth TableInputs: Four binary inputs, interpreted as a four-bit binary numberOutputs: Seven outputs, for each of the seven segments
number A B C D a b c d e f g0 0 0 0 0 1 1 1 1 1 1 01 0 0 0 1 0 1 1 0 0 0 02 0 0 1 0 1 1 0 1 1 0 13 0 0 1 1 1 1 1 1 0 0 14 0 1 0 0 0 1 1 0 0 1 15 0 1 0 1 1 0 1 1 0 1 16 0 1 1 0 1 0 1 1 1 1 17 0 1 1 1 1 1 1 0 0 0 08 1 0 0 0 1 1 1 1 1 1 19 1 0 0 1 1 1 1 1 0 1 110 1 0 1 0 x x x x x x x11 1 0 1 1 x x x x x x x12 1 1 0 0 x x x x x x x13 1 1 0 1 x x x x x x x14 1 1 1 0 x x x x x x x15 1 1 1 1 x x x x x x x
Invalid inputs, assume zero
segment a = A’B’C’D’ + A’B’CD’ + A’B’CD + A’BC’D + A’BCD’ + A’BCD + AB’C’D’ + AB’C’D (canonical SOP)
segment a = A’C + A’BD + AB’C’ + B’C’D’ (minimal SOP)
segment a = (A+B+C+D’) (A+B’+C+D) (canonical and minimal POS)
![Page 16: The Connection: Truth Tables to Functions](https://reader036.vdocuments.site/reader036/viewer/2022081420/568150f3550346895dbf121b/html5/thumbnails/16.jpg)
Circuits for Segment asegment a = A’C + A’BD + AB’C’ + B’C’D’ (minimal SOP)
segment a = (A+B+C+D’) (A+B’+C+D) (canonical and minimal POS)
A B C D
a
ABCD
AB
CD
a